Algebra and Tiling: Homomorphisms in the Service of Geometry

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Cambridge University Press, 1994 - Mathematics - 207 pages
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Often questions about tiling space or a polygon lead to questions concerning algebra. For instance, tiling by cubes raises questions about finite abelian groups. Tiling by triangles of equal areas soon involves Sperner's lemma from topology and valuations from algebra. The first six chapters of Algebra and Tiling form a self-contained treatment of these topics, beginning with Minkowski's conjecture about lattice tiling of Euclidean space by unit cubes, and concluding with Laczkowicz's recent work on tiling by similar triangles. The concluding chapter presents a simplified version of Rédei's theorem on finite abelian groups. Algebra and Tiling is accessible to undergraduate mathematics majors, as most of the tools necessary to read the book are found in standard upper level algebra courses, but teachers, researchers and professional mathematicians will find the book equally appealing.
  

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Contents

Chapter 2
35
Chapter 3
54
Chapter 4
85
Chapter 5
105
Chapter 6
126
Appendix C
143
Chapter 7
155
Epilog
187
Formal Sums
197
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About the author (1994)

Sherman Stein received his PhD from Columbia University. His research interests are primarily algebra and combinatorics. He has received the Lester R. Ford prize for exposition. He is now retired from teaching at the University of California, Davis.

Sandor Szabó received his PhD from Eötvös University. He currently teaches in the Institute of Mathematics and Informatics at the University of Pécs, in Hungary.

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